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Linearity

Linearity. 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity. Introduction (1).

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Linearity

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  1. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

  2. Introduction (1) • Nonlinearity causes intermodulation of two adjacent strongly interfering signals at the input of a receiver, which can corrupt the nearby (desired) weak signal we are trying to receive. • Nonlinearity in power amplifiers clips the large amplitude input. @ Modern wireless communications systems typically  have several dB of variation in instantaneous poweras a function of time require highly linear amplifiers

  3. Introduction (2) • SiGe HBTs exhibit excellent linearity insmall-signal (e.g., LNA)  large-signal (e.g.,PA) RF circuits despite their strong I-V and C-V nonlinearities • The overall circuit linearity strongly depends on  the interaction ( and potential cancellation) between the various I-V and C-V nonlinearities  the linear elements in the device : the source (and load) termination; feedback present • The response of a linear (dynamic) circuit is characterized by animpulse response function in the time domain  a linear transfer function in the frequency domain • For larger input signals, an active transistor circuit becomes a nonlinear dynamic system

  4. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

  5. Harmonics (1)

  6. Harmonics (2) • An “nth-order harmonic term” is proportional to An • HD2(second harmonic distortion) = / = ( neglect 3k3A3/4 term) • IHD2 ( the extrapolation of the output at 2ω and ω intersect) obtained by letting HD2 = 1 = 1  A = IHD2 = 2  IHD2 is independent of the input signal level (A)  HD2 = A / IHD2 ( one can calculate HD2 for small-signal input A )  OHD2 ( output level at the intercept point ), G (small-signal gain) OHD2 = G*IHD2 = k1*2 =

  7. Gain Compression and Expansion (1) • The small-signal gain is obtained by neglecting the harmonics.The small-signal gain ：k1The nonlinearity-induced term： 3k3A3/4 • As the signal amplitude A grows, becomes comparable to or even larger than k1A  the variation of gain changes with input  fundamental manifestation of nonlinearity • If k3 < 0, then 3k3A3/4 < 0  the gain decreases with increasing input level (A)  “gain compression” in many RF circuits  quantified by the “1 dB compression point,” or P1dB

  8. Gain Compression and Expansion (2) • The transformation between voltage and power involves a reference impedance, usually 50Ω. • Typically RF front-end amplifiers require -20 to –25 dBm input power at the 1dB compression point.

  9. Intermodulation (1) • A two-tone input voltage x(t) = Acosω1t +Acosω2t • The output has  not only harmonics ofω1 and ω2  but also “intermodution products” at 2ω1-ω2and 2ω2-ω1 (major concerns, close in frequency to ω1 and ω2 )

  10. Intermodulation (2) • Products output are given by • A 1-dB increase in the input results in  a 1-dB increase of fundamental output but a 3-dB increase of IM product • IM3 (third-order intermodulation distortion)

  11. Intermodulation (3) • IIP3 ( input third-order intercept point) is obtained by letting IM3 = 1  independent of the input signal level (A) IM3 can be calculated for desired small input AIM3 = A2 / IIP32 IIP3 can be measured by A0, IM30 IIP32 = A02 / IM30 • IIP3, A0  voltageIIP32, A02  power ( taking 10 log on both side ) 20 log IIP3 = 20 log A0 – 10 log IM30  PIIP3 = Pin + ½( Po,1st – Po,3rd )

  12. Intermodulation (4) • OIP3 = k1*IIP3  OIP32 = k12*IIP32 IIP32 = OIP32/ k12 = A2/IM32 OIP32 = (k1A)2/IM32 ( taking 10 log on both side )  20 log OIP3 = 20 log k1A – 10 log IM3  POIP3 = P o,1st + ½( Po,1st – P o,3rd)  The gain compression at very high input power level can be seen

  13. Intermodulation (5) • IIP3 is an important figure for front-end RF/microwave low-noise amplifiers, because they must contend with a variety of signals coming from the antenna. • IIP3 is a measure of the ability of a handset, not to “drop” a phone call in a crowded environment. • The dc power consumption must also be kept very low because the LNA continuously listening for transmitted signals and hence continuously draining power. • Linearity efficiency = IIP3 / Pdc( Pdc = the dc power dissipation )  excellent linearity efficiency above 10 for first generation HBTs  competitive with Ⅲ-Ⅴ technologies

  14. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

  15. Physical Nonlinearities in a SiGeHBT • ICE the collector current transported from the emitter the ICE-VBE nonlinearity is a nonlinear transconductance • IBE the hole injection into the emitter also a nonlinear function of VBE. • ICB the avalanche multiplication current a strong nonlinear function of both VBE and VCB has a 2-D nonlinearity because is has two controlling voltages. • CBE the EB junction capacitance  includes the diffusion capacitance and depletion capacitance a strong nonlinear function of VBE when the diffusion capacitance dominates, because diffusion charge is proportional to the ICE • CBC  the CB junction capacitance

  16. Equivalent circuit of the HBT

  17. The ICE Nonlinearity (1) • i(t) : the sum of the dc and ac currentsvc(t) : the ac voltage which controls the conductanceVC : the dc controlling (bias) voltage • For small vc(t), considering the first three terms of the power series is usually sufficient.

  18. The ICE Nonlinearity (2) • The ac current-voltage relation can be rewritteniac(t) = g vc(t) + K2g vc2(t) + K3g vc3(t) + …g : the small-signal transconductanceK2g : the second-order nonlinearity coefficientK3g : the third-order nonlinearity coefficient • For an ideal SiGe HBT, ICE increases exponentially with VBEICE = IS exp (qVBE/kT)

  19. The ICE Nonlinearity (3) • The nonlinear contributions to gm,eff increase with vbe. • Improve linearity by decreasing vbe.

  20. The IBE Nonlinearity • For a constant current gain βIBE = ICE/βgbe = gm/βK2gbe = K2gm/βK3gbe = K3gm/βKngbe = Kngm/β • For better accuracy, measured IBE-VBE data can be directly used in determining the nonlinearity coefficients.

  21. The ICB Nonlinearity (1) • The ICB term represents the impact ionization (avalanche multiplication) current ICB = ICE (M-1) = IC0(VBE)FEarly(M-1)IC0 : IC measured at zero VCBM : the avalanche multiplication factorFEarly : Early effect factor • In SiGe HBT, M is modeled using the empirical “Miller equation”  VCBO and m are two fitting parameters

  22. The ICB Nonlinearity (2) • At a given VCB, M is constant at low JC where fT and fmax are very low. • At higher JC of practical interest, M decreases with increasing JC, because of decreasing peak electric field in the CB junction (Kirk effect). • m, VCBO, ICO, VRare fitting parameters  also varies with VCB

  23. The ICB Nonlinearity (3) • The fT and fmax peaks occur near a JC of 1.0-2.0 mA/μm2, while M-1 starts to decrease at much smaller JC values. • ICB is controlled by two voltages, VBE(JC) and VCB2-D power series • iu = gu uc + K2gu uc2 + K3gu uc3 + …iv = gv vc + K2gv vc2 + K3gv vc3 + …iuv = K2gu&gv uc vc + K32gu&gv uc2 vc + K3gu&2gv uc vc2  cross-term

  24. The CBE and CBC Nonlinearity (1) • The charge storage associated with a nonlinear capacitor • The first-order, second-order, and third-order nonlinearity coefficients are defined as

  25. The CBE and CBC Nonlinearity (2) • qac(t) = C vc(t) + K2C vc2(t) + K3C vc3(t) + … • The excess minority carrier charge QD in a SiGe HBT is proportional to JC through the transit time τf QD = τf ICE = τf IS exp (qVBE/kT)

  26. The CBE and CBC Nonlinearity (3) • The EB and CB junction depletion capacitances are often modeled by C0, Vj, and mj are model parameters • The CB depletion capacitance is in general much smaller than the EB depletions capacitance. However, the CB depletion capacitance is important in determining linearity, because of its feedback function.

  27. The CBE and CBC Nonlinearity (4) • Caution must be exercised in identifying whether the absolute value or the derivative is dominant in determining the transistor overall linearity.

  28. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

  29. Volterra Series - Fundamental Concepts (1) • A general mathematical approach for solving systems of nonlinear integral and integral-differential equations. • An extension of the theory of linear systems to weakly nonlinear systems. • The response of a nonlinear system to an input x(t) is equal to the sum of the response of a series of transfer functions of different orders ( H1, H2, ……, Hn ).

  30. Volterra Series - Fundamental Concepts (2) • Time domain  hn (τ1, τ2,…., τn) is an impulse response Frequency domain  Hn ( s1, … , sn ) is the nth-order transfer function obtained through a multidimensional Laplace transform Hn takes n frequencies as the input, from s1=jω1 to sn=jωn • H1(s), the first-order transfer function, is essentially the transfer function of the small-signal linear circuit at dc bias. • Solving the output of a nonlinear circuit is equivalent to solving the Volterra series H1(s), H2(s1,s2), H3(s1, s2, s3),….

  31. Volterra Series - Fundamental Concepts (3) • To solve H1(s) the nonlinear circuit is first linearized solved at s = jω requires first-order derivatives • To solve H2(s1,s2),H3(s1,s2,s3) also need the second-order and third-order nonlinearity coefficients • The solution of Volterra series is a straightforward case the transfer functions can be solved in increasing order by repeatedly solving the same linear circuit using different excitation at each order

  32. First-Order Transfer Functions (1) • Consider a bipolar transistor amplifier with an RC source and an RL load • Neglect all of the nonlinear capacitance in the transistor, the base and emitter resistance, and the avalanche multiplication current • Base node “1”, Collector node “2” Y(s)  the admittance matrix at frequency s H1(s)  the vector of the first-order transfer functionI1(s)  a vector of excitations

  33. First-Order Transfer Functions (2) • By compact modified nodal analysis (CMNA) Fig 8.9 to Fig 8.10 • By Kirchoff’s current law  node 1  node 2

  34. First-Order Transfer Functions (3) • The corresponding matrix • For an input voltage of unity (Vs = 1) V1 and V2 become the transfer functions at node 1,2 • The firs subscript represents the order of the transfer function,and the second subscript represents the node numberH11,H12

  35. Second-Order Transfer Functions (1) • The so-called second-order “virtual nonlinear current sources” are applied to excite the circuit. • The circuit responses (nodal voltages) under these virtual excitations are the second-order transfer functions. • The virtual current source  placed in parallel with the corresponding linearized element  defined for two input frequencies, s1 and s2 determined by 1)second-order nonlinearity coefficients of the specific I-V nonlinearity in question determined by 2) the first-order transfer function of the controlling voltage(s)

  36. Second-Order Transfer Functions (2) • The second-order virtual current source for a I-V nonlinearityiNL2g(u) = K2g(u) H1u(s1) H1u(s2) K2g(u) : second-order nonlinearity coefficient that determines the second-order response of i to u H1u(s) : the first-order transfer function of the controlling voltage u

  37. Second-Order Transfer Functions (3) • iNL2gbe = K2gbe H11(s1) H11(s2)iNL2gm = K2gm H11(s1) H11(s2) • The controlling voltage vbe is equal to the voltage at node “1,” because the emitter is grounded. • The virtual current sources are used to excite the same linearized circuit, but at a frequency of s1 + s2.

  38. Second-Order Transfer Functions (4) • Y : CMNA admittance matrix at a frequency of s1 + s2H2 (s1,s2) : second-order transfer function vectorI2 : a linear combination of all the second-order nonlinear current sources, and can be obtained by applying Kirchoff’s law at each node • The admittance matrix remains the same, except for the evaluation frequency.

  39. Third-Order Transfer Functions (1) • Y : CMNA admittance matrix at a frequency of s1 + s2 + s3 • H3(s1,s2,s3) : the third-order transfer function • The third-order virtual current source for a I-V nonlinearityiNL3g(u) = K3g(u) H1u(s1) H1u(s2) H1u(s3)+2/3 K2g(u) [ H1u(s1) H2u(s2,s3) + H1u(s2) H2u(s1,s3) + H1u(s3) H2u(s1,s2) ] K2g(u)  the second-order nonlinearity coefficientK3g(u)  the third-order nonlinearity coefficientH1u(s)  the first-order transfer functionH2u(s1,s2)  the second-order transfer function

  40. Third-Order Transfer Functions (2) • iNL3gbe(u) = K3gbe(u) H11(s1) H11(s2) H11(s3)+2/3 K2gbe(u) [ H11(s1) H21(s2,s3) + H11(s2) H21(s1,s3) + H11(s3) H21(s1,s2) ] • iNL3gbe(u) = K3gbe(u) H11(s1) H11(s2) H11(s3) +2/3 K2gbe(u) [ H11(s1) H21(s2,s3) + H11(s2) H21(s1,s3) + H11(s3) H21(s1,s2) ]

  41. Linearity 8.1 Nonlinearity Concept 8.2 Physical Nonlinearities 8.3 Volterra Series 8.4 Single SiGe HBT Amplifier Linearity 8.5 Cascode LNA Linearity

  42. A Single HBT amplifier for Volterra series analysis

  43. Circuit Analysis • Y and I are obtained by applying the Kirchoff’s current law at every node. • IIP3 (third-order input intercept) at which the first-order and third-order signals have equal power • IIP3 is often expressed in dBm usingIIP3dBm = 10 log (103 IIP3)

  44. Distinguishing Individual Nonlinearities • The value that gives the lowest IIP3 (the highest distortion) can be identified as the dominant nonlinearity.

  45. Collector Current Dependence • For IC > 25mA, the overall IIP3 becomes limited and is approximately independent of IC. • Higher IC only increases power consumption, and does not improve the linearity.

  46. Collector Voltage Dependence (1) • The optimum IC is at the threshold value.

  47. Collector Voltage Dependence (2)

  48. Load Dependence (1) • The load dependence results from the CB feedback, due to the CB capacitance CCB and the avalanche multiplication current ICB. • Collector-substrate capacitance (CCS) nonlinearity since VCS is a function of the load condition

  49. Load Dependence (2) • CCB = 0, ICB = 0, note that IIP3 becomes virtually independent of load condition for all of the nonlinearities except for the CCS nonlinearity.

  50. Dominant Nonlinearity Versus Bias • ICB and CCB nonlinearities are the dominant factors for most of the bias currents and voltages. • Both ICB and CCB nonlinearities can be decreased by reducing the collector doping. • But high collector doping suppresses Kirk effect.

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